(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Men Of Mathematics"

MEN  OF MATHEMATICS
In sharp distinction to this is the other phrase which is reiter-
ated over and over again in mathematical writing: 'There
exists.' For example, some would say 'there exists a whole
number greater than 2', or 'there exists a number less than n
and greater than n  2.' The use of such phraseology definitely
commits its user to the creed which Kronecker held to be
untenable, unless, of course, the 'existence1 is proved by a con-
struction. The existence is not proved for the sets (as defined
above) which appear in Cantor's theory.
These two ways of speaking divide mathematicians into two
types: the *we can' men believe (possibly subconsciously) that
mathematics is a purely human invention; the * there exists'
men believe that mathematics has an extra-human 'existence'
of its own, and that *we' merely come upon the 'eternal truths'
of mathematics in our journey through life, in much the same
way that a man taking a walk in a city comes across a number
of streets with whose planning he had nothing whatever to do.
Theologians are 'exist' men; cautious sceptics for the most
part 'we' men. 'There exist an infinity of even numbers, or of
primes', say the advocates of extra-human 'existence'; 'produce
them', say Kronecker and the 4we* men.
That the distinction is not trivial can be seen from a famous
instance of it in the New Testament. Christ asserted that the
Father 'exists'; Philip demanded 'Show us the Father and it
sufficeth us.' Cantor's theory is almost wholly on the 'existence'
side. Is it possible that Cantor's passion for theology deter-
mined his allegiance? If so, we shall have to explain why
Kronecker, also a connoisseur of Christian theology, was the
rabid *we' man that he was. As in all such questions ammunition
for either side can be filched from any pocket.
A striking and important instance of the 'existence' way of
looking at the theory of sets is afforded by what is known as
Zermelo's postulate (stated in 1904). 'For every set M whose
elements are sets P (that is, M is a set of sets, or a class of
classes), the sets P being non-empty and non-overlapping (no
two contain things in common), there exists at least one set fi
which contains precisely one element from each of the sets P
which constitute M.' Comparison of this with the previously
632